ANNs & RNNs can be used to create some great models in many different domains, including time-series forecasting. However, across all of these domains, they suffer from the problem of hyper-parameter optimization. Because neural networks are so flexible, it is not clear, at the outset, which arrangement of neurons will be most effective to solve a given problem. It is also not clear how fast the network should learn from new signals, what sorts of activation functions to use in the different layers of the network, and which of several possible regularization methods might be best. Making these decisions well requires either years of practice and experience, or a lot of trial and error (or, maybe both!).
In contrast, a regression-based method like ARMA will typically have just a couple of simple hyperparameters, each of which has a clear, intuitive, meaning. This means that an untrained practitioner can probably get an ARMA result that is close to the result of a trained practitioner using ARMA.
Essentially: neural networks are brittle and sensitive to the choice of hyper-parameters, while regression generally is not.